Question

Relationship between \(\mathbf{r}\) and \(\mathbf{r}^{\prime}\) Consider the helix \(\mathbf{r}(t)=\langle\cos t, \sin t, t\rangle,\) for \(-\infty

Short answer

Answer: The position vector \(\mathbf{r}\) and its first derivative \(\mathbf{r}^{\prime}\) are orthogonal only at the point \(\langle 1, 0, 0\rangle\) on the helix.

Step-by-step solution

Calculate the first derivative

Differentiate the position vector with respect to \(t\). \(\mathbf{r}(t)=\langle\cos t, \sin t, t\rangle\). \(\mathbf{r}^{\prime}(t)=\frac{\mathrm{d}\mathbf{r}(t)}{\mathrm{d}t}=\langle-\sin t, \cos t, 1\rangle\).

Calculate the dot product of \(\mathbf{r}(t)\) and \(\mathbf{r}^{\prime}(t)\)

Next, we need to find the dot product of the position vector and its derivative. \(\mathbf{r}(t)\cdot\mathbf{r}^{\prime}(t) = (\cos t)(-\sin t)+(\sin t)(\cos t)+(t)(1)\).

Solve the equation for when the dot product is zero

We need to find \(t\) values when the dot product is zero. $\mathbf{r}(t)\cdot\mathbf{r}^{\prime}(t) = -\sin t\cos t+\sin t\cos t+t=0\\ t=0$ Note that since the helix is an infinitely long curve extending in both directions, \(\sin t\) and \(\cos t\) are always nonzero but they cancel each other out in the expression we derived. In this case, the expression simplifies to \(t=0\), meaning the position vector and its derivative are orthogonal only at \(t=0\).

Find the corresponding points on the helix

Finally, we need to find the corresponding point on the helix when \(t=0\). \(\mathbf{r}(0) = \langle\cos(0), \sin(0), 0\rangle = \langle 1, 0, 0\rangle\). Hence, the position vector \(\mathbf{r}\) and its first derivative \(\mathbf{r}^{\prime}\) are orthogonal only at the point \(\langle 1, 0, 0\rangle\) on the helix.

Key concepts

Helix

A helix is a fascinating three-dimensional curve that often resembles a spring or a spiral staircase. It occurs when an object moves in a circular motion while progressing along a straight line. Unlike a flat circle, a helix has a component of movement in an additional dimension, typically height. For example, a common mathematical helix might look like:

• Moving horizontally around a circle (defined by its radius).
• Simultaneously moving vertically (along an axis, like height).

This creates a smooth and seamless spiraling pathway. The helix equation \(\mathbf{r}(t)=\langle\cos t, \sin t, t\rangle\) describes such a curve, where:

• \(\cos t\) and \(\sin t\) represent the circular horizontal motion, creating a circle in the xy-plane.
• \(t\) represents the vertical motion, showing the rise along the z-axis as time progresses.

This combination of rotational and linear motion defines a helix and gives it its characteristic shape.

Orthogonal Vectors

Vectors are orthogonal if they meet at a right angle (90°) to each other, which means their dot product is zero. In simpler terms, they are perpendicular, similar to how the x-axis and y-axis are aligned in perpendicular directions on a graph. This is a key concept in vector calculus because it helps to identify angles and relationships between vectors.For our problem, we need to find when the tangent vector \(\mathbf{r}^{\prime}(t)=\langle-\sin t, \cos t, 1\rangle\) is orthogonal to the position vector \(\mathbf{r}(t)=\langle\cos t, \sin t, t\rangle\). When these are orthogonal, their dot product equals zero, suggesting that they form a right angle at that point. Finding this point is essential as it gives us the specific location on the helix where the vectors meet at right angles.

Dot Product

The dot product is a very useful tool in vector calculations that tells us about the angle between two vectors. It also gives information on the projection of one vector over another. Mathematically, the dot product of two vectors \(\mathbf{a} = \langle a_1, a_2, a_3 \rangle \) and \(\mathbf{b} = \langle b_1, b_2, b_3 \rangle \) is given by: \[ \mathbf{a}\cdot\mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \]In our problem, the dot product is used as: \[ \mathbf{r}(t)\cdot\mathbf{r}^{\prime}(t) = \cos t \times (-\sin t) + \sin t \times \cos t + t \times 1 \]We look for when this result equals zero to find where these two vectors are orthogonal. By solving the expression and finding \( t = 0 \), we determine the exact moment on the helix when these vectors are perpendicular. This utilizes the property of the dot product being zero to solve real-world geometric relations.